# What is the right way to set up two random tensor fields which have an identical average diffusivity

I want to compare some properties of traveling waves through two randomly diffusive media. The traveling waves follow the fisher equation:

$$\frac{du}{dt} = \nabla(\mathbf{D}_{\gamma} \nabla u) + u(1-u)$$.

I am creating a random field of diagonal matrices in the following form:

$$\mathbf{D}_{\gamma}(\mathbf{x}) = \mathbb{1 \cdot \gamma}$$

Here, $$\gamma$$ is an equally distributed random value within a certain range centered around 1.0.

e.g.:

$$\gamma \in (0.8,1.2)$$

$$\gamma \in (0.5,1.5)$$

I am interested in the distribution of the gradient magnitudes vs the field amplitudes, so basically $$\nabla u (u)$$. I run two simulations for a different spread of diffusivities and the traveling wave builds up nicely in both cases. They also have approximately the same speed of advance and general shape.

The wave-front is progressing from the bottom left outwards. I can now plot for every degree of freedom what its amplitude is (x-axis), and what the gradient is (y-axis):

As expected, the spread of the relationship of $$\nabla u$$ vs $$u$$ is larger where the diffusive tensor are scaled on a wider range.

How do I make sure that the average diffusivity is indeed comparable when choosing a higher range for $$\gamma$$? I am currently taking the naive approach that as the range is centered around 1.0, the average diffusivity of the two compared fields should be comparable too. I am worried that there might be some pitfalls of statistics I am unaware of. I am also worried that this approach would leave the simulations to be resolution-dependent.

How do I make sure that the two random material property fields are comparable while increasing their 'spread'? Or in other words, what kind of statistical property do I have to enforce in the two tensor fields to compare the two runs? Is it the average determinant, average trace...Frobenius norm etc.?

• To check if I am understanding, $D$ is a diagonal tensor with entries that are random (not equal) and each point in the domain is different (in general) but follows a particular distribution? Aug 2 '19 at 18:24
• currently I use $\mathbb{D} = \mathbb{1} ~ \gamma$. So all diagonal entries are identical per cell, but vary from point to point throughout the domain. I am open to other suggestions, but that seemed as the easiest choice for a start. I want to ensure that the randomly perturbed case and the isotropic case stay comparable. Aug 5 '19 at 7:15
• Are you thinking on homogenization or a multi-scale analysis of some sort maybe? Aug 6 '19 at 23:34

## 1 Answer

There is no reason to believe that two random fields with the same arithmetic mean would yield solutions that have anything to do with each other. In fact, for the case you consider, one might imagine that maybe the harmonic mean is actually a better indicator, but even that is unclear -- it could also be the geometric mean.

Apart from this, you have to actually say anything about the spatial randomness. A coefficient where $$\gamma$$ is a Gaussian field with a short correlation lengthscale is going to yield solutions that are going to look very different than if the spatial correlation is long. Both are going to be very different from ones where $$\gamma$$ is spatially discontinuous. In other words, randomness comes in many different ways that go far beyond just specifying an interval.

• Thank you for your answer, I extended my question a bit to explain what I want to do. On large enough scales the behaviour of the wave-front should only depend on the average diffusivity (or some other statistical property). I want to investigate the relationship between the gradient vs amplitude of the wave-front's profile. Qualitatively I already see that the spread in effective diffusivities results in a spread around a comparatively simple relationship (a polynomial), but I want to make sure that the these two cases indeed are comparable. Aug 2 '19 at 8:14
• I also want to add, that as of now, I only compare the two runs on identical resolutions. Aug 2 '19 at 8:23
• But you haven't addressed my main issue: "Randomness" is not only determined by the distribution of values at each point, but also by spatial correlations. Aug 4 '19 at 11:29
• Okay. So in the example above I only compare the isotropic case and the "randomized" case on identical resolutions. If I were to refine the grid, I agree that the correlation length would shrink with it, and it is to be expected that that would have an effect. Aug 5 '19 at 7:20
• Correct. Right now, your coefficient is correlated on length scales of the mesh size, and uncorrelated beyond that. In the limit of $h\rightarrow 0$, you will end up with an $L^\infty$ randomly distributed coefficient. I see no reason to believe that the limit of the solution as $h\rightarrow 0$ exists, or is in any sense meaningful. Aug 5 '19 at 23:32